Problem 4 - Discrete Mathematics with Applications

Question

In Exercises \(1-6, a_{n}\) denotes the \(n\) th term of a number sequence satisfying the given initial condition(s) and the recurrence relation. Compute the first four terms of the sequence. $$ \begin{array}{l}{a_{1}=1, a_{2}=2} \\ {a_{n}=a_{n-1}+a_{n-2}, n \geq 3}\end{array} $$

Step-by-Step Solution

Answer

The first four terms of the number sequence are \(1, 2, 3, 5\).

Step 1: Find the first term (a_1)

The first term of the sequence, denoted as \(a_1\), is given in the problem as \(a_1 = 1\). So, the first term of the sequence is 1.

Step 2: Find the second term (a_2)

Similarly, the second term of the sequence is given in the problem as \(a_2 = 2\). So, the second term of the sequence is 2.

Step 3: Find the third term (a_3)

To find the third term (\(a_3\)), we will use the given recurrence relation, which states that for \(n \geq 3\), \(a_n = a_{n-1} + a_{n-2}\). Since we are looking for the third term, we plug in \(n= 3\): \(a_3 = a_{3-1} + a_{3-2} = a_2 + a_1\). Using the previously found values of \(a_1 = 1\) and \(a_2 = 2\), we can calculate \(a_3 = 1 + 2 = 3\). So, the third term of the sequence is 3.

Step 4: Find the fourth term (a_4)

To find the fourth term (\(a_4\)), we will again use the given recurrence relation. Plugging in \(n= 4\), we get: \(a_4 = a_{4-1} + a_{4-2} = a_3 + a_2\). Using the previously found values of \(a_2 = 2\) and \(a_3 = 3\), we can calculate \(a_4 = 3 + 2 = 5\). So, the fourth term of the sequence is 5. In conclusion, the first four terms of the sequence are \(1, 2, 3, 5\).

Key Concepts

Fibonacci sequence

The Fibonacci sequence is a series of numbers where each number is the sum of the two preceding ones. Specifically, it starts with the numbers 0 and 1. However, in different contexts, such as the exercise you encountered, it might start with different numbers like 1 and 2, depending on the initial conditions provided. Generally, the sequence can be expressed using the formula:

• \( F_1 = 0 \)
• \( F_2 = 1 \)
• \( F_n = F_{n-1} + F_{n-2} \) for \( n \geq 3 \)

This sequence has fascinating properties and appears in numerous natural phenomena, such as the arrangement of leaves on a stem, the flower petal count, and much more. In your exercise, you've seen a version of a Fibonacci-like sequence that adapts the same concept. The fun part about learning these sequences is how they demonstrate a simple yet powerful concept of progression.

number sequences

A number sequence is a list of numbers arranged in a specific order. Each number in the list is termed an element or term of the sequence. Number sequences are often used in various mathematical concepts to illustrate patterns or rules. The rule that defines the sequence may involve simple linear patterns, geometric patterns, or more complex relations such as those found in the Fibonacci sequence. In your exercise, the numbers are formed using a recurrence relation. This is a way of defining sequences where each term is a function of its preceding terms. It helps to understand how sequences evolve and is a foundational concept in understanding more complex mathematical patterns. Recurrence relations can model a variety of numerical phenomena and help you predict future terms without listing all the previous ones. An essential part of working with sequences is identifying the pattern or rule behind the arrangement of the numbers. This involves looking at initial terms (given explicitly) to compute future terms.

initial conditions

Initial conditions in the context of number sequences specify the starting terms required before applying a recurrence relation. They are a crucial part of understanding how a sequence begins and progresses. Without initial conditions, a recurrence relation alone is insufficient to determine the exact sequence of numbers. In your exercise, the initial conditions are \( a_1 = 1 \) and \( a_2 = 2 \). These conditions set the stage for the entire sequence, providing the necessary starting points for calculating subsequent terms using the recurrence relation \( a_n = a_{n-1} + a_{n-2} \) for \( n \geq 3 \). Think of initial conditions like the first dominoes in a row—pushing them creates a chain reaction, much like how the first few terms of a sequence determine all subsequent terms. Recognizing and correctly applying these conditions is key to accurately extending the sequence as needed.